How would you integrate xlnx using Risch algorithm (I know how to use integration by parts)?

Question

I know integration by parts, that's not what I'm interested in. Risch is a very advanced integration algorithm that I'm trying to get to grips with.

Answer 1

Check out:

http://www.math.colostate.edu/~hulpke/lectures/m676ca/maplerisch.pdf

as well as the excellent paper by Moses that you can read online if your school library pays for the rights:

http://portal.acm.org/citation.cfm?doid=362637.362651

If not, the citation is:

Moses, Joel
Symbolic integration: The stormy decade.
Communications of the ACM 14 (1971), 548--560.

Answer 2

You are looking for a function F whose derivative is x ln x. Now this logarithm, very generally, comes either from a function involving 1/x or from a function involving a power of a logarithm. Suppose a term in the solution F has the form

F = u * (ln v)^n

where u, v are functions of x. The derivative would be

F' = u' * (ln v)^n + n*u*(v'/v)*(ln v)^(n-1)

Since x ln x only involves a first power of ln x, we must choose n = 1. The formula suggests that we take u'(x) = x and v(x) = x, so we try

F(x) = (x^2 ln x)/2

The derivative is

F'(x) = x ln x + x^2/2*(1/x)*(ln x)^0
= x ln x + x/2

which is almost what we are looking for -- the term x/2 is easy to remove:

F(x) = (x^2 ln x)/2 - x^2/4